Theorem termco 49971

Description: The object of a terminal category. (Contributed by Zhi Wang, 17-Nov-2025.)

Hypotheses

Ref	Expression
termcbas.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcbas.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
termco	⊢ (𝜑 → ∪ 𝐵 ∈ 𝐵)

Proof of Theorem termco

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		termcbas.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2		termcbas.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3	1, 2	termcbas 49970	. 2 ⊢ (𝜑 → ∃𝑥 𝐵 = {𝑥})
4		unieq 4849	. . . . . 6 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 = ∪ {𝑥})
5		unisnv 4858	. . . . . 6 ⊢ ∪ {𝑥} = 𝑥
6	4, 5	eqtrdi 2790	. . . . 5 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 = 𝑥)
7		vsnid 4595	. . . . 5 ⊢ 𝑥 ∈ {𝑥}
8	6, 7	eqeltrdi 2847	. . . 4 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 ∈ {𝑥})
9		id 22	. . . 4 ⊢ (𝐵 = {𝑥} → 𝐵 = {𝑥})
10	8, 9	eleqtrrd 2842	. . 3 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 ∈ 𝐵)
11	10	exlimiv 1937	. 2 ⊢ (∃𝑥 𝐵 = {𝑥} → ∪ 𝐵 ∈ 𝐵)
12	3, 11	syl 17	1 ⊢ (𝜑 → ∪ 𝐵 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {csn 4555  ∪ cuni 4838  ‘cfv 6485  Basecbs 17170  TermCatctermc 49962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-termc 49963

This theorem is referenced by: (None)